Theorem rspcegf 39496

Description: A version of rspcev 3340 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
rspcegf.1	⊢ Ⅎ𝑥𝜓
rspcegf.2	⊢ Ⅎ𝑥𝐴
rspcegf.3	⊢ Ⅎ𝑥𝐵
rspcegf.4	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
rspcegf	⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Proof of Theorem rspcegf

Step	Hyp	Ref	Expression
1		rspcegf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
2		rspcegf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐵
3	1, 2	nfel 2806	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
4		rspcegf.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
5	3, 4	nfan 1868	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓)
6		eleq1 2718	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
7		rspcegf.4	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
8	6, 7	anbi12d 747	. . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓)))
9	1, 5, 8	spcegf 3320	. . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)))
10	9	anabsi5 875	. 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
11		df-rex 2947	. 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))
12	10, 11	sylibr 224	1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523  ∃wex 1744  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780  ∃wrex 2942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233

This theorem is referenced by:  rspcef  39555  stoweidlem46  40581